Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems

Abstract

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For β > 1 let T β be the β-transformation on [0, 1]. We determine the Lebesgue measure and Hausdorff dimension of the setfor infinitely many n ∈ N .If in addition τ 1 (x), τ 2 (y) < 1 for all x, y ∈ [0, 1] and max y∈[0,1] τ 2 (y) < log β2 β 1 , then the Hausdorff dimension of the setfor infinitely many n ∈ N is also determined, where g 1 , g 2 : [0, 1] 2 → [0, 1] are two Lipschitz functions.